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A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

%I A353844 #5 May 28 2022 16:37:07
%S A353844 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,25,27,29,31,32,37,40,41,43,47,
%T A353844 49,53,59,61,63,64,67,71,73,79,81,83,84,89,97,101,103,107,109,112,113,
%U A353844 121,125,127,128,131,137,139,144,149,151,157,163,167,169,173,179
%N A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.
%C A353844 The run-sums transformation is described by Kimberling at A237685 and A237750.
%C A353844 The runs of a sequence are its maximal consecutive constant subsequences. For example, the runs of {1,1,1,2,2,3,4} are {1,1,1}, {2,2}, {3}, {4}, with sums {3,3,4,4}.
%C A353844 Note that the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so this sequence lists Heinz numbers of partitions whose run-sum trajectory reaches an empty set or singleton.
%e A353844 The terms together with their prime indices begin:
%e A353844       1: {}            25: {3,3}           64: {1,1,1,1,1,1}
%e A353844       2: {1}           27: {2,2,2}         67: {19}
%e A353844       3: {2}           29: {10}            71: {20}
%e A353844       4: {1,1}         31: {11}            73: {21}
%e A353844       5: {3}           32: {1,1,1,1,1}     79: {22}
%e A353844       7: {4}           37: {12}            81: {2,2,2,2}
%e A353844       8: {1,1,1}       40: {1,1,1,3}       83: {23}
%e A353844       9: {2,2}         41: {13}            84: {1,1,2,4}
%e A353844      11: {5}           43: {14}            89: {24}
%e A353844      12: {1,1,2}       47: {15}            97: {25}
%e A353844      13: {6}           49: {4,4}          101: {26}
%e A353844      16: {1,1,1,1}     53: {16}           103: {27}
%e A353844      17: {7}           59: {17}           107: {28}
%e A353844      19: {8}           61: {18}           109: {29}
%e A353844      23: {9}           63: {2,2,4}        112: {1,1,1,1,4}
%e A353844 The trajectory 60 -> 45 -> 35 ends in a nonprime number 35, so 60 is not in the sequence.
%e A353844 The trajectory 84 -> 63 -> 49 -> 19 ends in a prime number 19, so 84 is in the sequence.
%t A353844 ope[n_]:=Times@@Prime/@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k];
%t A353844 Select[Range[100],#==1||PrimeQ[NestWhile[ope,#,!SquareFreeQ[#]&]]&]
%Y A353844 This sequence is a subset of A300273, counted by A275870.
%Y A353844 The version for compositions is A353857, counted by A353847.
%Y A353844 A001222 counts prime factors, distinct A001221.
%Y A353844 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353844 A124010 gives prime signature, sorted A118914.
%Y A353844 A304442 counts partitions with all equal run-sums.
%Y A353844 A353851 counts compositions with all equal run-sums, ranked by A353848.
%Y A353844 A325268 counts partitions by omicron, rank statistic A304465.
%Y A353844 A353832 represents the operation of taking run-sums of a partition.
%Y A353844 A353833 ranks partitions with all equal run-sums, nonprime A353834.
%Y A353844 A353835 counts distinct run-sums of prime indices, weak A353861.
%Y A353844 A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A353844 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353844 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353844 A353866 ranks rucksack partitions, counted by A353864.
%Y A353844 Cf. A005811, A073093, A130091, A181819, A182857, A304660, A325239, A325277, A353839, A353862, A353867.
%K A353844 nonn
%O A353844 1,2
%A A353844 _Gus Wiseman_, May 26 2022